134 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS series which are at first rapidly convergent, and which enable us to calculate the numerical values of the integrals with extreme facility. These expressions were first given by M. Cauchy, in the case of Fresnel's integrals, to which the integrals just written arc equivalent. They may readily be obtaiiied by integration by parts, though it is not thus that they were demonstrated by M. Cauchy. If now the above expressions be substituted for the integrals in (195) the terms containing P destroy each other, and for general values of t the most important term after the first contains £"*. Since however t is supposed to correspond to the end of an oscillation, so that nt is a multiple of ir, the coefficient of this term vanishes, and the most important term that actually remains contains only t~%. Hence neglecting insensible quantities we get from (195) (100). We get from (194) by performing the integrations log ° ----9 — c / / gin nt (cos ?&£ + sin nt) (it A. 'v An J o = ~ ./^~ {2nt +1 - cos 2-/i< - sin 2/^J, . MI V 2?il J which becomes since nt is a multiple of TT log^ + *4»« /^.^...............(1<)7). 0 J. 4>n v 2'^' We get from (196) and (197) A, >10H-A/1,, A whence log , A. .(IDS), and the same relation exists between the common logarithms of the arcs, which are proportional to the Napierian logarithms. Now LogJ.0 — Log J. is the quantity immediately deduced from experiment, and Log(^l0 + A^10)— Log Alt is the correction to be applied, in consequence of the circumstance that the motion began from rest. Instead of applying the proportionate correct-ion